Calculus, 8th Edition,stewart; page 49 problem 1

Question

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes.

(a) If P is the point (15, 250) on the graph of V, find the slopes of the secant lines PQ when Q is the point on the graph with t = 5, 10, 20, 25, and 30.

(b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines.

(c) Use a graph of the function to estimate the slope of the tangent line at P. (This slope represents the rate at which the water is flowing from the tank after 15 minutes.)

Answer 1

Step:1

A tank holds  gallons of water.

(a)

The point is .

Find the slopes of the secant lines .

Consider the point .

At  and the corresponding the value of  is .

So the point is .

Slope of the two points is .

The slope of the secant line  is .

At  and the corresponding the value of  is .

So the point is .

The slope of the secant line  is .

At  and the corresponding the value of  is .

So the point is .

The slope of the secant line  is .

At  and the corresponding the value of  is .

So the point is .

The slope of the secant line  is .

At  and the corresponding the value of  is .

So the point is .

The slope of the secant line  is .

The slopes of the secant lines  are  and .

Step:2

(b)

Find the average of the slopes of the secant lines near to .

Consider the points are near to .

Points are  and .

The slopes of the secant lines  are formed from the points  and  is  and .

The average of the slopes is 

Therefore the slope of the tangent line at  is .

Comments

Step:3

(c)

Graph :

Use the values from the table and graph the function.

(1) Draw the coordinate plane.

(2) Plot the points from the table.

(3) Connect the plotted points to a smooth curve.

(4) Draw a approximate tangent line at .

From the graph, the green line represents the approximate tangent line at .

So the slope of the tangent line is .

Solution:

(a) The slopes of the secant lines  are  and .

(b) The slope of the tangent line at  is .

(c) The slope of the tangent line after  minutes is .